Writing a book from the beginning to the end is (so I heard) a very hard process. Planning a book is easier. This question is dual in a sense to the question "Books you would like to read (if somebody would just write them)". It is about a book that you feel you would like to write (if you just have the time). A book that will describe a topic not yet properly discribed or give a new angle to a subject that you can contribute.

The question is meant to refer to realistic or semi-realistic projects (related to mathematics). Answers about book projects based on existing survey articles or lecture notes can be especially useful.

Of course, If you had some progress in writing a book mentioned here please please update your answer!

Dear Daniel, No, this is about a book you feel you are capable of writing (perhaps more so than anybody else) but you just dont have yet the time or energy to do it. For example, look at Alan Hatcher's book projects math.cornell.edu/~hatcher/#anchor1772800 . This site describes nice future book projects. (In this case, these books are likely to be written sometime in the future.) So the question was about ideas/projects of a similar kind.
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Gil KalaiFeb 3 '11 at 20:25

@Gerhard: Did you mean that you'd like to write a book on System Design? :)
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J.C. OttemFeb 3 '11 at 22:09

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Gerhard, I like to ask you, if I may, why do you put the phrase "Ask Me About System Design" between your first and last name.
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Gil KalaiFeb 3 '11 at 22:11

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Gerhard, maybe it will be useful if you added to your user page some link to your page or an email so people who are interested can contact you. Not being particularly fornd of advertisements, the habit of putting some advertisemnt item between the first and last name does not come accross to me as a good habit. (But I suppose we got used to it by now).
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Gil KalaiFeb 3 '11 at 22:44

Why: Because they are one of the first examples of spaces that are almost, but not quite, entirely unlike manifolds. They are relatively straightforward spaces which can be fairly conceptually grasped, but still contain enough intricacies to reveal some of the important differences between finite and infinite dimensions (though perhaps I should say between manifolds modelled on Banach spaces and more general manifolds). A book on their differential topology would thus be a gentle introduction to the topic than is (as far as I'm aware) currently available (in particular, although just about everything I'd want to say is covered in Kriegl and Michor's works, it's in such a context and with such generality that "daunting" doesn't quite cut the mustard).

Who For: Me, 10 years ago. That is, I'd try to write the book I wish I'd had when starting out in infinite dimensional differential topology so I wouldn't have made all the mistakes that I made.

Why Me: Because I work in that area and I think I've made just about every wrong assumption about loop spaces possible so I know lots of the traps for unwary differential topologists venturing out into the miasma that is infinite dimensional topology.

Will I Ever Actually Write It: Maybe, maybe not (vote for this answer if you want me to!). I made a start by writing up some seminar notes. I've started transferring them in to the nLab (but in the process I've been generalising them which slightly goes against the purpose of the project as I described it above). I'd certainly like to write it, if only to convince myself that I no longer have all those false assumptions, but whether or not I ever actually do it ... (hey, I've an idea, maybe all the time I put into MO and meta.MO could be reallocated to book-writing. Then it'll be finished next week.).

Question seems a little silly to me, unless it's meant as motivation. But for those who answer the question and then are motivated to go ahead with their book project, I can offer some personal experience on the process.

Step 1. Start with a detailed outline and 100+ pages of detailed notes from a course that you've taught on the subject.

Step 2. Estimate about how long you think it will take to turn those notes into a published book. (In my case, I figured that it couldn't take more than a year or so.)

Hi Angelo, more generally I was referring to realistic or semirealistic book projects. (So a research monograph with a proof of the GRH would also not be considered as a good answer.) anyway, why is your answer an answer rather than a comment?
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Gil KalaiFeb 4 '11 at 18:23

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To Gil: Well, it was an answer, even if not a serious one. Anyway, it was a joke, I did not mean it to be offensive (even if it is true that I do not think it's a good question).
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AngeloFeb 4 '11 at 19:35

I would love to write something titled "Higher mathematics from engineer's perspective", which would consist of a few chapters each of which should be devoted to a single simple to state real engineering problem whose solution requires rather sophisticated mathematical tools. The main content of the chapter would be the shortest path to the full solution with all the relevant concepts explained, all relevant theorems fully proven, etc. For instance, one possible such chapter would be "How to shape an airplane wing and compute the lifting force?" with all that complex analysis etc. An easier one would be "How to shape the rollercoaster track?" about elementary space curve theory. A harder one would be "How to find defects in solids?" with PDE's, wave equation, etc. Something like that is certainly lacking though I doubt that the people who will read it need it and that the people who need it will read it.

Another book written by a mathematician that makes me really jealous is "Alice in Wonderland". Alas, I currently do not have any good idea of how to beat it though the perception of the surrounding reality by a mathematically inclined mind is much more subtle and "unusual" today than it was in Carroll's time. (I almost wrote "perverted" instead of "unusual" but it is a kind of "perversion" that is in the reality itself, not in its perception, so this word, if used, won't really be understood correctly without a long explanation).

Needless to say, I will write neither of the two. Still, somewhere in the Platonic domain both these books exist and occasionally I stumble upon an "excerpt" that is taken right from one of them (that "excerpt" is, of course, not necessarily in the form of a written text or a sound track, but I cannot find a better word (fragment?) now).

I guess an engineer might object to the way you describe your first book: "How to find defects in solids?", for instance, has a lot of different engineering answers, each of which probably has its own mathematics (unless you want to remain extremely vague about the mathematical details). Thus, more appropriate titles for one of your chapters might be "how to find surface defects using ultraviolet rays", for instance.
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Thierry ZellFeb 4 '11 at 19:04

Sure. Once you come to the actual writing (which will never happen in this case), you make the things like titles etc. more precise. My only goal was to show that such a book won't be empty and the level of difficulty may vary quite a bit. The whole point here is not to be "vague" about mathematical details, but, on the contrary, to be as precise as possible and to present all relevant mathematics rigorously. There are chapters where I know next to nothing myself but I know people who actually worked on related projects personally, etc.
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fedjaFeb 4 '11 at 21:07

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@fedja: when you actually write your book (i.e., never), here is something to consider: based on the title alone I would never read it, but based on your description I would very much like to read it. This makes me think that the title is somehow inaccurate.
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Pete L. ClarkFeb 5 '11 at 0:54

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The title is accurate. The titles of numerous written glossy textbooks named "Calculus for ...", "Applied ...", "Basics of ..." aren't.
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fedjaFeb 5 '11 at 20:30

While I find the question borderline, I succumb to the temptation to answer.

Knot Theory: Kawaii examples for topological machines.

Topology is full of big machines, which may seem rather daunting to the student. But knot theory is a wonderful playground for toy models of many of these machines, where you can see how they work and visualize what they are doing. And one can draw pictures.
I think that a collection of these examples would be useful to students (I would have loved to have had it) or to people who would like to teach topology. And I don't think anything like this exists, really. The machine itself would be introduced only briefly, refering to somewhere else for more detail, while the knot theory example would be fleshed out in full.
For example, curvature of knots is the perfect playground for the Gauss-Bonnet Theorem. Computations of homology in knot theory give perfect toy examples (with pictures you can draw) for Mayer-Vietoris, the snake lemma, and other homological arguments. Ideas such as localization and Brown representability come up naturally. And an Alexander module gives a perfect playground for commutative algebra over a UFD.
So the idea would be to give sophisticated proofs of simple facts, letting the topological machines play the lead role. The student of topological machine X might then read the book by looking up the relevant section, which would give a kawaii (cute?) example in knot theory, highlighting how exactly the machine is working, and shedding light on its nature.
How likely am I to write it? I've toyed with the idea for a long time. For the book to be useful, it needs to be very visual and pedagogical, to make it light fluffy reading for one who knows the machine, and educational reading for one who doesn't. And becaue I have high asprations for it, it may take a while. But I do have intentions of actually writing it at some point, even if I don't yet know when that might be.

For "Kawaii", see e.g. en.wikipedia.org/wiki/Cuteness_in_Japanese_culture I don't know a good English translation- "cute" doesn't capture the meaning at all. The Hebrew "chamud" is much closer, as in "raayon chamud". Note that Wakimoto describes sl(2,C) as "Kawaii" in "Infinite Dimensional Lie Algebras", which in the English translation is rendered "charm".
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Daniel MoskovichFeb 3 '11 at 22:51

Book Title:An Introduction to Forcing (for people who don't care about foundations.)

Synopsis: Forcing is one of the most amazing techniques in use today, and it offers amazing insight into how objects in mathematics can be constructed. The aim of this book would be to focus on the tools and methods of Forcing, and provide examples of constructions which highlight the intrinsic beauty that can be found hiding under the surface of a forcing argument. Moreover, it would highlight the practical applications of, and sense of naturalness the "Forcing Perspective" brings to inductive mathematical constructions (which might be outside the domain of set-theoretic interest.)

Reason For Wanting to Write It: When I first learned about Forcing, the first thing that struck me was "Why the hell has no one ever told me about this? What the hell!? This is AWESOME!" That sense of awe has stayed with me throughout my very short "career." So the book would be a way for me to share this view with other mathematicians who don't really care all that much about "set theory", "category theory", or "foundations" (just like I did before I learned about independence proofs, etc.) Moreover, the aim would not be to convert them to some relativist view of mathematics, but to just show them how directly linking the logical structure of an object with its construction can open new doors, and add much needed perspective to any field.

When Would It Get Written: Honestly, not now, and not in the near future, maybe 10/20 years. The reason for this is, I just don't know enough yet, I'm still a student. That being said, I must admit, I am most likely not the first person anyone would pick to write such a book. However, if I was ever presented with the opportunity I would take it in a heartbeat. To me the importance of the ideas and perspective for mathematics as a whole out weigh the possible huge list of errors and corrections that would follow such a book (if written by me that is).

PS: if there are any spelling or grammar errors, feel free to fix them.

Basically, I'd like to collect together the stuff I mentioned in this answer, as well as some more. The information is scattered through many books and papers right now. Many of these papers are very challenging, even though they often contain elementary parts that could stand on their own.

"Thinking with categories" a small introduction for the layman.
May be a more commercial title would be "Functorial Thinking".
A small book (circa 120 p.) with the goal of explaining basic category theory using plenty of examples but mostly non mathematical ones.
Intended for an audience of linguists, philosophers, computer designers and any curious intellectual.

The book presuppose a reader not adverse to a minimum of algebra, yet it should mostly contains basic defining algebraic equations for categories, functors , natural transformations and adjunctions.

The goal of this book: It should enable a philosopher (not necessarily specialized in logic) to grasp properly what an adjunction is in 2 to 4 hours.

To illustrate : A 5-subset of a football team can be made by picking some players randomly, but a sub-object is a set of 5 players that can play together! In fact common language would call it sub-team.
So far when trying to design examples in real life you end up too often with groupoids and thin category(posets).

Any suggestions of places from which to draw material/inspiration would be most welcome.

@Qiaochu : Yes I have read this Lawere book , but it almost always mathematical examples sets that are real-world illustrated. Here the aim is to describe categorically real-world situations.
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Jérôme JEAN-CHARLESFeb 7 '11 at 4:05

Lawvere and Schanuel did have some real-world things, for example a very nice example about Chinese restaurants...
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Todd Trimble♦Feb 21 '12 at 22:32

The idea would be to give a complete account of the interaction of these two central set-theoretic concepts, aiming at their intersection, rather than at their union. How are large cardinals affected by forcing? What kinds of forcing can preserve which kinds of larger cardinals? To what extent do the standard forcing notions affect large cardinals? For example, to what extent can we preserve large cardinals while forcing GCH, V=HOD, or their negations, among other set-theoretic features commonly obtained by forcing? By what methods can we show that large cardinals are preserved? To what extent can large cardinals be made indestructible by (certain kinds of) forcing? What are the most general things that can be said about how the large cardinal embeddings of one model relate to the large cardinal embeddings of its forcing extensions and ground models? The topic has a fundamentally category-theoretic flavor, since it is at essence about how large cardinal embeddings are affected by forcing, ideas that can often be expressed by means of large commutative diagrams, involving lifts of embeddings from models to their forcing extensions.

Let me confess: the truth is that I have been working on writing such a book for the past ten years, and have about 320 pages completed, sitting on my computer; I have used drafts of this book when teaching graduate courses in set theory, and over the years I have allowed various versions of these drafts to become distributed to various other researchers. In fact, it appears that this "book" has already been cited a number of times by various authors in published articles, even though it does not yet exist as a book.

So I would like to complete it. But somehow I keep getting distracted by other interesting and worthwhile projects...

I'd like to read/write a book on constructible sheaves and the six operation formalism on complex analytic stacks, as it seems there are not too many references in literature (I would apologize if there is one that I'm not aware of), and there are so many basic facts in étale cohomology that one expects (at least I expect, in my research) to be true for analytic stacks but I couldn't find any reference, and therefore had to prove them from the beginning.

Over the years I had a few ideas about books as well as the appealing idea of not writing a book. When I see books others have written I am usually quite amazed by them, and the amount of work involved seems rather alarming. (Being able to write unpolished things and to jump from one topic to another is an advantage of writing a blog.) In any case, I would prefer to write a book with an electronic version using the full possibilities of hyperlinks. Here are some specific ideas about books I would have liked to write had this been painless:

1) Face numbers, graphs and skeleta of polytopes and complexes. This is an area of combinatorial geometry which I find very exciting and it is related to various other areas of combinatorics and mathematics. (I am quite an expert in the area of the proposed book but not an expert in these related areas.) This topic is discusses in several books but I don't think there is a book devoted to this subject. My starting idea for this project is simple: To take Chapters 18 (by Billera and Bjorner) and Chapter 20 (by me) from the Handbook of discrete and computational geometry update them and add proofs.

2) Analysis of Boolean functions. This is a fairly new research area which again I find very exciting. It has connections to various areas of combinatorics and computer science, to probability and to harmonic analysis. Yet it is a sufficiently young field that a book is possible. How to go about it? Muli Safra and I wrote a related survey article about thresold phenomenon what seems to be missing is an additional survey on Fourier analysis of Boolean functions and then adding-proofs transformation as part of what is required to make them into a book.

3) A different idea that Gunter Ziegler and I played with was to write " The book of examples" (mimmicking perhaps the style of Aigner and Ziegler's "Proofs from the book") The mathoverflow question on fundamental examples is very much related to this idea. So given the many answers all that is "left to do" is to select some of the examles, to divide them into chapters, to ellaborate more on each selected example and indicate important connections. (This can also be done collectively.)

4) A different direction would be to transform the posts from my blog "Combinatorics and More" into a book (like Terry Tao and Dick Lipton have done for their blogs.)

5) (ADDED: AUG 2011) I forgot to mention that I did write an Internet Book Entitled "Gina Says: Adventures in the Blogosphere String War" , which contains all sort of things and also some mathematics. I would like to edit it further to make it suitable to a larger audience and possibly publish it via a commercial publisher.

6) (ADDED: Nov 2012) The content of my debate with Aram Harrow on quantum fault-tolerance that started in this post and concluded in this post over the blog "Godel lost letter and P=NP" can be the basis for an interesting book.

To modify advice a colleague once gave me: The decision to write (another) book is like the decision to have (another) child: the work increases exponentially, but the rewards are commensurate with the effort. I hope you do write one or more of these wonderful books you've outlined!
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Joseph O'RourkeFeb 13 '11 at 15:46

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Thanks a lot, Joe! I do not think that with books the work grows exponentially with the number of books (but you have infinitely more experience, is it exponential?), but rather the burden of work grows exponentially with the inverse of the remaining work needed to complete the project.
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Gil KalaiFeb 13 '11 at 16:33

Serge Lang would roll his eyes at us trying to formulate a psychological book burden-of-work law! :-)
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Joseph O'RourkeFeb 13 '11 at 17:40

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Gil, regarding the immense work associated to finishing a project, in Chapter 24 of Alexander Solzhenitzyn's novel the "The First Circle", the mathematician Sologdin describes this vividly in terms of "the final inch". Solzhenitzyn's essay is much-cited in the medical literature, see for example the PubMed listing for "Dracunculiasis eradication: the final inch" (here dracunculiasis is the Guinea worm disease). Yes, translating math-to-medicine is a lot of work.
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John SidlesFeb 25 '11 at 9:26

It seems that the point of most calculus books currently existing is to: 1. Provide homework problems 2. Provide sample solutions to such problems so that students can pattern match 3. provide formulas in little boxes.

I would like to write a calculus book which really forces students to think about calculus. This means that they will have to develop the calculus themselves. The book will assist in this task by asking very leading questions, and asking students to work out examples which contain the essence of each new idea. A course based on such a book would consist of students working through the relevant section the night before, and the "lecture" is a group discussion aimed at clarifying the ideas developed. Of course, this must be supplemented with plenty of calculations, but these must always be accompanied by written explanations of the thought process behind each calculation.

Of course, these thoughts apply equally well to any other book about mathematics, especially those aimed at undergraduates. The ones aimed at graduate students or researchers could also benefit from this, but by that time most students have learned how to do this kind of thing for themselves.

@Steven: This sounds very much like a modified Moore method class. There are some well-developed and peer-reviewed notes of this style in the Journal of Inquiry Based Learning online.
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Carl MummertJul 24 '11 at 12:18

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Steven,check out Charles McCluer's HONORS CALCULUS It's basically a theoretical calculus course done in a Moore method style. It's too terse for my liking,but it's written very well with a lot of physical applications (!) Also,McCluer encourages instructors to develop their own courses with as much or as little detail as possible from it.This is a great challenge I'd love to try one day and it may serve as the skeleton for the kind of course you're suggesting.
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Andrew LJul 24 '11 at 17:30

In an ideal world where I would have unlimited time for nice book projects, I would like to write an update and english translation of my book on Poisson geometry and deformation quantization, which is unfortunately in german (I was young, needed the money...)

In course of such an update and translation I would like to incorporate some new topics (any suggestions?) and include in particular a treatment of symmetries, Morita theory, and existence&classification of star products also for the Poisson case (based on formality and gloalization a la Dolgushev...), and perhaps, also some more details on reduction. On the other hand, I would try to make the symplectic and Poisson geometry part much shorter, perhaps even in form of an appendix, to focus on the DQ part. I would like to keep the balance between mathematical presentation of the material with additional motivation section from mathematical physics.

But the world is far from being ideal, so I can not promise when I will find time for doing so... ;)

What this country needs is a successor to Courant/Robbins' "What is mathematics?", first published in 1941. Gowers' wonderful "Princeton companion to mathematics" cannot serve as a modern replacement of this volume, insofar as it addresses a group which is already deeply interested in mathematics and definitely knows what mathematics is all about. Not unlike Gowers' compendium the book I'm dreaming of would be the work of a devoted collective of authors, but in addition it would need a unifying editorship to make it the landmark in the field for decennia to come, as it was the case with Courant/Robbins' book seventy years ago.

After reflexion, I think I will reduce my contribution to this: Don't think too much about the book you want to write, just write it down. Don't wait that everything is perfect, just begin. Anyway, it will take years.

-1 While I find this answer mildly amusing, in a certain sense, I would very much like to see this question staying open. At the moment this seems not a given, and answers of this form might well support the case for a closure.
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quidFeb 3 '11 at 20:49

I would like to write a book about algebraic shifting. Your survey is too compact and (in my opinion) is not user-friendly. On the other hand rewriting all proofs with algebraic machineries kills the beauty of this theory (again, in my opinion). So I don't like Herzog and Hibi's book. I think it is necessary to show the concrete combinatorial nature of algebraic shifting.

It is hardly polite to criticize an author's paper in a public forum! I suggest you reword or stay silent.
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Joseph O'RourkeFeb 21 '12 at 20:08

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I did not mean to criticize Kalai's article. I should mention that Kalai invented algebraic shifting theory. However by the last sentence I meant the original works of Gil Kalai! Thank you for your suggestion.
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Afshin GoodarziFeb 22 '12 at 19:02

I am very flattered, Af1323, that you want to write a book abour algebraic shifting!
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Gil KalaiMar 1 '12 at 17:48

"The Laws of Relations" the title echoing famous "The Laws of Thought" By George Boole but in spirit closer to "Logic of Relatives" by Charles Sanders Peirce. The subject is algebra of relations with named attributes of arbitrary finite arity. It is predicate calculus without quantifiers where the predicates are identified by their names only (not the names, nor positions of predicate attributes) with syntax reminiscent of Peirce-Tarski Relation algebra.

Admittedly, the number of publications on the subject is less than dozen, which makes it pie-in-the-sky sort of wish.

It would be based on a course I took 2 years ago at the CUNY Graduate Center given by John Terilla and Tom Tradler and would focus on the basic concepts of the Gerstenhaber bracket, deformations of associative algebras,operads and quantum groups.It would differ from the usual texts in that would have very minimal prequisites:strong undergraduate backgrounds in algebra and topology (an algebra course based on Herstien and a topology course based on John Lee's book would suffice).

It would focus mainly on the material in "classical" deformation theory i.e. Gerstenhaber's original papers as well as the work done by Stasheff and Markl on the theory. It would end concievably with a glimpse at the modern theory on sheaves and prepare the student for Harshorne's book and the current literature.

I'm not really sure this is a suitable answer to your question, but I'd like to have my recently published novel "Sahelios", in which a Japanese highschool student named Satori (Japanese female given name standing for "awakening") proves a generalized version of the Riemann Hypothesis, translated into English. Had I been truly bilingual, I would have written it directly in English. The good thing with a novel is that you can formulate ideas in an easier to understand fashion, and hopefully make new people with no prior knowledge of number theory get interested in the subject.

Apart from that, I'd like to write a detailed mathematical lexicon for researchers in several languages, say French, English, Spanish, German and Japanese, covering rather extensively the subjects that a graduate student is supposed to know and focusing on number theory, algebraic geometry, representation theory, category theory and functional analysis. This could be very useful for graduate students or established researchers dealing with languages they don't master enough to do math in the considered languages, but I strongly doubt I'll ever have the courage to write such a book.

Why Write This Book? The ideas of Gil and Aram and Scott --- about "Computers that work a different way, and why maybe we can't build them" --- are beautiful thoughts, and so we can hope that the team of all three of them is right, and that much good comes of their fine ideas. This book hopes to give some short, simple thoughts of how their good ideas might be made more nearly real for everyone, so that we arrive at good places by ways that are more fast and sure.

Al's Happy Thoughts The way that we tell this story begins with a well-known number-teacher named Al (who still lives today). Al loves simple ideas, and what he writes reminds us of something that all of us already know from our own lives:

"New number-pictures in our brains can make hard problems seem soft and easy! Each new picture is like a little wave of water, that slowly lifts our minds above even our most-hard problems, until even little children can see how to fix them easily."

Al kept busy during every hour of the first half his of life to build these beautiful new number-pictures and share them.

Al's More Serious (and Sad) Thoughts Then during the second half of his life, Al began to think very hard about a more serious set of problems.

"Our world will soon have one hundred-hundred-hundred-hundred-hundred people living on it. The people of this huge number need safe homes and good jobs, so that they have chances to smile and hug and marry each other, and make more happy families, and do good works for each other. But instead, people hate and fight each other --- and even entire states hate and fight each other --- so that many people are killed, and very many more are hurt."

Mean-while, the world it-self has become slowly more sick, year-by-year from the huge crowds of people who live on it.

The Heavy Hearts Of Young People These hard facts give a heavy heart to everyone who lives today. Young people (especially) are right to wonder, "Will we ever be happy, and find hope, and even find a nice man/woman to love and start our own family? And when will it happen --- if ever --- that people and states leave behind hate and fear and hurt and killing?"

Al thought very hard about these heavy problems for many tens of years, but yet in his long struggle he found no good answers, and so (slowly) he has become old and sad --- even crazy, some say --- and so now-a-days Al hides himself and talks to no-one (not even his old friends, when they come to look for him).

Ideas Full of Hope However! During these same tens of years, more-and-more people are coming to understand that the beautiful number-ideas and number-pictures that came from the first half of Al's life can help practically to fix the hard problems that have filled the rest of Al's life. The way this works is as follows:

To begin, it is especially nice that the problem that Gil and Aram and Scott all love to think about --- "computers that would work in a different way" (as Scott says) even if we have to "draw a different picture [which] is hard and takes a lot of time" (as Gil says) --- gives people a beautiful (and fun!) way to quickly bring many of Al's number-ideas to life and especially, find practical answers to the hard questions that made him so sad.

The Too-Big Block of Books Everyone knows that building "computers that would work in a different way" isn't easy. To carefully read through a two-foot block of hard books --- having to understand more than ten-hundred-hundred single number-facts along the way! --- would be barely enough for a young person to even begin to think about how to build a practical one.

Al's Work Gives Us Hope We can take heart from Al's work, though. His number-pictures and number-ideas let us pack the key ideas of the whole two-foot block of "different computer" books into just seven short "Green Pieces" (together with a short end-story about what the number-pictures mean to doctors and hospitals, and a few other pictures too). This helps young people (especially) see that the big block of books need not be read as hundreds of hard-to-understand and hard-to-even-remember stories, but rather can be read as one easy-to-understand story that surprises us by being rather short-and-simple. This is a reason for every-one to say "Thank you!" to Al (and friends like Emmy who help Al to think-up and share these beautiful, practical ideas).

Al's Work Makes Gil's Ideas Real A beautiful thing about Al's ideas is that they help make Gil's deep ideas both stronger and more real. Here the short, simple story is this: once Gil starts us with a set of key number-truths, then Al's number-ideas immediately help us draw a number-picture of a world in which Gil's set of number-truths becomes real. This amazes us! Because as our number-thoughts begin to run along this new "Al-to-Gil" number-trail, we find over-and-over that looking into "Al's spaces" in search of "Gil's truths" shows us more number-surprises even than looking into "David's spaces" to build the "different computers" Aram and Scott like so much.

"Different Computers" Are Good for NOTHING -- And That's Great! Next, we notice a nice fact about Scott's "different computer" --- these computers are not (at present) good for any practical work what-so-ever. And so, when we have good number-ideas (along the lines of Al and Gil), then there is no reason at all to hide those number-ideas ... instead, it's definitely a good idea to share those ideas fast. Here there is no reason to whisper, and every reason to shout!

Let's Look Ahead To Better Times When we look more far-ahead in time, we see yet another good reason to share these number-ideas for free ... a reason that some people with well-known names (like Si and Johnny and Norbert and Dick) were the first to state:

"We say to our doctor friends: You should use more-and-better number-pictures (like the story of Al and Gil and Aram and Scott makes us want to learn). Especially, to help doctors learn faster, you should find out how to make the look-at-tiny-matter bit-box focus better. Then it will become very easy to learn how sick people can be helped to become well: we can just look at their living matter-bits! Once our doctors can focus their look-at-tiny-matter bit-box one hundred times stronger, then the hard problems of helping sick people to become whole-and-well will be made very much easier."

When Bad News is Good News As we learn new number-reasons why "different computers" are hard to make, those same new number-reasons help us to find new and better ways to make the focus of our look-at-tiny-matter bit-box one hundred times stronger. As Dick reminds us (and Si and Norbert and Johnny too): "That will be GREAT!"

So Let's Go Fast! Helping sick people is a very strong reason to share new number-ideas fast, for a reason that everyone knows, but makes us sad when we speak it out loud. For too many tens-of-years, from too-many states all around the world, mothers and fathers have sent their young sons and daughters away to fight, and many of these young people have come back not to home, but to a hospital. Perhaps a pretty good start at fixing Al's heavy problem-set can be made, by using Al's own number-ideas to find faster, better ways to make whole-and-well all those people who were sent to fight.

It's Hard At First, But Then It Gets Easy Many people find Al's number-ideas and number-pictures are hard to understand, but everyone understands that these ideas and pictures can help sick people become whole-and-well. And more: everyone wants the better jobs and new business start-ups that grow-up all around us, as we work together to help sick people to become whole-and-well. That is how everyone share, with a good heart, in this great world-wide work.

This big picture gives each of us good reasons to want to put in the long hours, and walk the long trails, and do the hard jobs, and beat the many doubts, that we need to realize the many practical uses of Al's number-ideas!

True Words from the Old Days That is why the true words that a number-person named Jean said almost two-hundred years ago, still sound good to our minds and hearts today:

Reason For Hope This biggest of all, most beautiful of all, most hope-full work of all --- a great work in which we all share, that helps make people whole-and-well, so they can leave hospitals and return to normal lives --- can join both the hearts and the business interests of every state, as a world-wide job that (especially) lifts-up the hopes of those mothers and fathers and family, who sent their young people to fight in causes that so often seem good in the beginning.

This is how (already!) we all began to live a story that --- as the world hopes --- will have a best-ever and nicest-ever real-world ending.

A Huge "Thank You" Thank you Gil and Aram and Scott, for your free sharing of your many fun, beautiful, important ideas with all the world!

Hmmm ... the strongly bimodal distribution of responses to this book-preface are bringing home to me the lessons that Joseph Landsberg's essay Clash of Cultures describes in the preface to his (wonderful) book Tensors: Geometry and Applications (2012).
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John SidlesDec 27 '13 at 21:01

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@JohnSidles If by responses to the book preface you mean votes, I have a suspicion that the negative reception might be partly due to the large number of edits that continually bring this to the top of the MO stack. Historically this has been something that irritates people.
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Todd Trimble♦Dec 27 '13 at 22:20

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@ToddTimble Ouch! I had no notion of this MathOverflow dynamic, and my sincere apologies are tendered to everyone. Especially, Todd, thank you very much for telling me this ... I've been having colleagues and family read the essay aloud, then editing per each person's comments, not knowing that this was exactly the wrong thing to do. The contrast between the favorable readings and the unfavorable ratings now is explained, and so this preface will now lay fallow till after the holidays. Again, sincere apologies are extended to everyone.
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John SidlesDec 27 '13 at 22:32

Prologue Here is an excerpt from Gowers'
comment on Gödel's Lost Letter:

Gowers' choice “If on the other hand P!=NP, then the price I ask …
is that we come to understand far better the subclass of
mathematical statements and proofs we are actually interested
in. … I would like a world where exactly one of the
statements ‘P=NP’ and ‘mathematical
creativity can be automated’ is true.”

Let us regard Gowers' choice (as we will call it) not as a wish, but as an
engineering directive whose fulfillment requires a bespoke
mathematical toolset such that “creativity can be automated.” Specifically, we regard Gowers' choice as a path toward Bill Thurston's goal:

Thurston's goal “The goal of mathematics is to develop enhanced ways for humans to see and think about the world. Mathematics is a transforming journey, and progress in it can better be measured by changes in how we think than by the external truths we discover.

Lincoln's objective “To make mutual exchange [of] discovery, information, and knowledge; so that, at the end, all may know every thing, which may have been known to but one, or to but a few [and] to stimulate that discovery and invention into extraordinary activity.”

Balancing these various ideas, we design the mathematical formalisms of the Gowers-Thurston world with a view toward providing “enhanced ways for humans to see and think about” their individual participatory roles in the emerging “extraordinary activities” of the 21st century.

Schwinger's template “Although ‘1’ is not perfectly
‘0’ we can effectively regard …”

Applying Schwinger's template, we “effectively regard” the mathematical
toolset of Gowers' choice and Thurston's goal as arising from this ansatz:

The Gowers-Thurston-Schwinger (GTS) Ansatz “Although ‘NP’ is not known to be formally
separable from ‘P’ we can effectively regard it as
such whenever our main purpose is mathematical understanding.
Similarly, although ‘Hilbert space’ is not known to
be perfectly a ‘low-dimension secant variety of a Segre variety’
we can effectively regard it as such whenever our main purpose
is dynamical understanding.”

The first part of the GTS ansatz restricts NP (and thus P) to those algorithms whose runtime attributes are
decidable and whose outputs (including random samples) are verifiable; Juris Hartmanis has suggested that this restriction (suitably formalized) might render P and NP provably separable. In effect, the ansatz restricts P and NP to those algorithms that are humanly understandable in the Gowers-Thurston sense. The second
part of the GTS ansatz focuses upon systems (both classical and quantum) whose trajectories are dynamically compressed onto low-dimension algebraic manifolds. In effect, the ansatz restricts computational simulations to the noisy and/or low-energy and/or highly symmetric dynamical trajectories that are commonly encountered in nature, in technology, and in the laboratory.

Needless to say, the MathSTEMnet reviews are entirely imaginary; in particular, the review of Volume III seeks to retell a classic Robert Heinlein medical narrative from 1958 in the dryly arch mathematical voice of Joseph Doob's 1948 review of Claude Shannon's Mathematical theory of communication (MR0026286).

This volume aims to provide solid foundations for classical
and quantum simulation. In the first of its three parts students
learn the basics of differential and algebraic geometry at the
same time that they learn the basics of Hamiltonian dynamics,
first in the context of classical molecular dynamics, then in the
context of classical interacting spins. From the beginning all
state-spaces are treated as algebraic varieties (specifically,
secant varieties of Segre varieties) that are endowed with
symplectic and metric structure. The second of three parts treats
(classical) thermostats and (quantum) Lindbladian
processes within a mathematically natural Hamiltonian/Stratonovich
formalism. In the final
part, classical and quantum tools are merged in the practical
context of quantum spin biomicroscopy, viewed both as a Shannon
communication channel and as a target for simulation and sensing
in synthetic biology.

The resulting volume reads as though Saunders Mac Lane,
Vladimir Arnold, and Joe Harris teamed up to cover in one volume
the dynamical elements of three classic texts: (1) Charlie
Slichter's Principles of Magnetic Resonance,
(2) Nielsen and Chuang's Quantum Computation and Quantum
Information and (3) Frenkel and Smit's
Understanding Molecular Simulation: from Algorithms
to Applications — all in the flowing
example-filled style of Jack Lee's Introduction to Smooth
Manifolds. It is suitable for a senior undergraduate or
first-year graduate course (that requires students
to unlearn some of what they previously have been
taught).

Volume II in this series takes up where Volume I leaves off:
with the description of the molecular dynamics and quantum spin
imaging of biological molecules. The first of three parts
surveys the quantum theory of spin polarization transport,
with an emphasis on transport-based techniques for
generating order-unity dynamic nuclear polarization (T-DNP). Substantial
emphasis is placed on efficient iterative evaluation of
“musical” isomorphisms in trajectory integrations.
The second part discusses 3D imaging methods that are enabled by
the coherent polarization so achieved. The third part discusses
the “crossover region”of imaging at 0.5 nm
resolution, below which molecular dynamical simulations carry
more information than direct imaging. Each chapter is accompanied
by two-part design exercises, the first consisting of a
pencil-and-paper (or SymPy) symbolic analysis, the second
consisting of a large-scale (SAGE/PyQSE) numerical simulation;
working code is provided for most exercises.

The concluding chapter requires students to design an
enterprise for spin-imaging the entire nucleus of a eukaryotic
cell (via quantum spin microscopy) at 0.5 nm resolution, then
refining that imaging information (via molecular simulation) to
sub-Angstrom scales. Present rapid developments in quantum
spin microscopy, sample hyperpolarization, and molecular dynamic
simulation ensure that this section
will be outdated within a very few years …and yet no
book better conveys the mathematical toolset that is so greatly
in-demand to support the burgeoning global enterprise of observational
synthetic biology.

It is now ten years since Volume I of this series appeared,
heralding a new era of comprehensive quantum spin imaging of
biomolecular structure, and comprehensive simulation of the the
molecular dynamics of these structures. It is now five years
since Volume II heralded a new era of synoptic information
regarding the workings of “every atom in its place”,
very much as von Neumann and Feynman foresaw last century. Now
Volume III has appeared, and the authors promise to provide a
mathematical “natural” toolset for applying these
capabilities in healing and regeneration.

Authors Ella Pomfrey and Finn Longbotton are members of the
new breed of physician that are comfortable with symplectic
structure and with bone structure, with individual molecules and
with individual patients, with genetic and epigenetic variation,
with complexity theory and with the
evolving cognition of healing brains. They have mastered, both
abstractly and in practice, the geometrically, algebraically,
combinatorically, and informatically natural tools that previous
generation of mathematicians brought to bear in the microscopic
theory of healing and regeneration. Now in this volume, Pomfrey
and Longbottom seek to bring this same natural toolset to bear on
macroscopic healing processes. The emphasis throughout is upon
practical clinical verification and validation procedures that
ensure that bone, nerves, and minds all cleave to a path that
leads to a satisfactory healing.

This reviewer entertains some doubt as to whether our
understanding of healing and regeneration, in particular
their epigenetic aspects, can ever match the naturality of our
microscopic understanding … but no-one is better
qualified than the authors, who have a distinguished record
in the regenerative treatment of battle trauma, to meet the
21st century's grand challenge of healing, by evolving a
mathematically natural understanding of it.

Thanks for the answer, John. At the end I found myself confused about the proposed books that you would like to write. (There was a different MO question about books you would like to see written.)
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Gil KalaiJul 22 '11 at 8:24

@Gil, thank you for your comment, and the post has been edited to clarify these points. Broadly speaking, the materials are in-hand to write Volume I, and if various experiments and algorithms work as planned, the writing of Volume II can commence in a few months. But Volume III is different---it will be written a decade from now (we hope not later) by people who have an MD/PhD skill-set that doesn't yet exist, who are informed by data-sets that don't yet exist, on the basis of clinical experience that doesn't yet exist. Yet it is the most important volume, and so we hope that day comes soon!
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John SidlesJul 22 '11 at 13:00

Perhaps I should mention too, that the reviewers Caradoc Dearborn, Dilys Derwent, and Mungo Bonham appear in the Harry Potter books as healers and/or Order of the Phoenix members. This is deliberate, as is the sequence A/B, C/D, E/F of authors' names (Alice/Bob, Carla/Dave, Ella/Finn): the purpose is to facilitate fan/fiction, per the Godel's Lost Letter essay "Time Chunks and Theory Nuggets" ( rjlipton.wordpress.com/2011/07/12/… ). Thus associated to these three math books is an imagined future history, to which I may post a link someday.
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John SidlesJul 22 '11 at 14:04

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Dear John, Am I correct in saying that: 1) The acronym STEM stands (here) for science, technology, engineering, and mathematics 2) The three books published by mathSTEMnet described at the last part of the answer are, in fact, three books you would like to write. A question: What do you mean by "Volume I exists today"?
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Gil KalaiJul 23 '11 at 8:44

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I did not vote on your answer (either way), but since you ask, I find so many edits, because of the resulting pumps of the question, a bit annoying. Perhaps some people simply downvoted as they where (also) annoyed by this.
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quidAug 1 '11 at 12:17